Is Northern Ireland the poison pill of hard Brexit? The visit of the new Irish prime minister, Leo Varadkar, to Belfast today is remarkable. This is not just for the astonishing sight of a southern politician who believes passionately in gay rights visiting the still conservative north – given how long the south’s reactionary Catholicism has been butt of northern ridicule. The visit is also part of Varadkar’s campaign to exploit Brexit as a tool of unification. The north-south border is one of the three “starter” issues of British EU withdrawal, to be resolved before a post-Brexit deal can be discussed.

Northern Ireland’s Democratic Unionists hold two contradictory positions: that hard Brexit is good; and that the border with the south must remain “porous”, for goods and people. They want tighter control on immigration into the UK, but know perfectly well that the border with the south cannot be closed. It is another case of wanting Brexit – “but not for me”.

Various versions of squaring this circle have been mooted, none satisfactory. A possible deal might involve tagging lorries heading north or south, requiring awesome bureaucracy and enforcement. People movement is harder. The prospect of Northern Ireland as a Balkan underbelly of EU movement into Britain may be exaggerated, but it is real.

Northern Ireland could remain de facto within the EU visa area, but with a formal passport border set up between Belfast and Britain. Dublin would love that, and Belfast hate it. Or Dublin could somehow become part of an all-UK passport zone, vetting travellers from the EU suspected of wanting to travel north. That would look like Ireland going semi-detached from the EU. Or nothing could happen, and the old cross-border rackets could resume. Having no border with the south would mean that EU migrants could continue flocking to Britain, albeit by a circuitous route. The British negotiators would not accept this.

To Irish republicans, Brexit is a golden opportunity for pushing unification of an island that has, like Cyprus, been a tragic casualty of identity politics for almost a century. An open border post-Brexit and a gradual merging of the subsidy-dependent north with the free-booting south makes every kind of sense. If Britain wants to “go it alone”, let the island of Ireland have no part in it. Let Europe’s border be the Irish Sea. Let Ireland unite.

If that prospect is too much for northern unionists to stomach, they had better get working. Hard Brexit is not just bad for the UK; in their case it is unworkable. Additional border controls around Northern Ireland – wherever drawn – would further isolate the province’s struggling economy. The DUP should fight for a single market and an open border. For once, it is in a parliamentary position to make a British government listen to it. But time is short.